Fast convergent series are presented for lattice sums associated with the simple cubic, face-centered cubic, body-centered cubic, and hexagonal close-packed structures for interactions described by an inverse power expansion in terms of the distances between the lattice points, such as the extended Lennard-Jones potential. These lattice sums belong to a class of slowly convergent series, and their exact evaluation is related to the well-known number-theoretical problem of finding the number of representations of an integer as a sum of three squares. We review and analyze this field in some detail and use various techniques such as the decomposition of the Epstein zeta function introduced by Terras or the van der Hoff–Benson expansion to evaluate lattice sums in three dimensions to computer precision.
Analytical formulas are derived for the zero-point vibrational energy and anharmonicity corrections of the cohesive energy and the mode Grüneisen parameter within the Einstein model for the cubic lattices (sc, bcc, and fcc) and for the hexagonal close-packed structure. This extends the work done by Lennard-Jones and Ingham in 1924, Corner in 1939, and Wallace in 1965. The formulas are based on the description of two-body energy contributions by an inverse power expansion (extended Lennard-Jones potential). These make use of three-dimensional lattice sums, which can be transformed to fast converging series and accurately determined by various expansion techniques. We apply these new lattice sum expressions to the rare gas solids and discuss associated critical points. The derived formulas give qualitative but nevertheless deep insight into vibrational effects in solids from the lightest (helium) to the heaviest rare gas element (oganesson), both presenting special cases because of strong quantum effects for the former and strong relativistic effects for the latter.
A smooth path of rearrangement from the body-centered cubic (bcc) to the face-centered cubic (fcc) lattice is obtained by introducing a single parameter to cuboidal lattice vectors. As a result, we obtain analytical expressions in terms of lattice sums for the cohesive energy. This is described by a Lennard-Jones (LJ) interaction potential and the sticky hard sphere (SHS) model with an r −n long-range attractive term. These lattice sums are evaluated to computer precision by expansions in terms of a fast converging series of Bessel functions. Applying the whole range of lattice parameters for the SHS and LJ potentials demonstrates that the bcc phase is unstable (or at best metastable) toward distortion into the fcc phase. Even if more accurate potentials are used, such as the extended LJ potential for argon or chromium, the bcc phase remains unstable. This strongly indicates that the appearance of a low temperature bcc phase for several elements in the periodic table is due to higher than two-body forces in atomic interactions.
The stability of the body-centered cubic (bcc) compared with the face-centered cubic (fcc) phase at finite pressures is investigated through exact lattice summations using a general (a, b) Lennard-Jones potential (a > b > 3). At zero pressure, the bcc phase is unstable or, at best, metastable for unphysical low values of the exponents (a, b) of the Lennard-Jones potential. From Helmholtz free energy calculations, we demonstrate that the stability of the bcc phase decreases with increasing pressure, with the metastable phase persisting into the highpressure range up to a high pressure limit at exponent a = 7.6603891 for the repulsive wall. The transition path is chosen to be of Bain type, connecting smoothly the two phases through a series of body-centered tetragonal (cuboidal) lattices.
We introduce two convergent series expansions (direct and recursive) in terms of Bessel functions and the number of representations of an integer as a sum of squares for N -dimensional Madelung constants, M N ( s ) , where s is the exponent of the Madelung series (usually chosen as s = 1 / 2 ). The convergence of the Bessel function expansion is discussed in detail. Values for M N ( s ) for s = 1 2 , 3 2 , 3 and 6 for dimension up to N = 20 are presented. This work extends Zucker’s original analysis on N -dimensional Madelung constants for even dimensions up to N = 8 .
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